3.1 Introduction #

1. Give an informal description of a set
2. Explain Russell’s Paradox

3.2 Sets and Set Operations #

1. Explain what a set is. How do we define a set? Can a set contain the same element twice?
2. When are two sets equal?
3. Give the definition of $A \subseteq B$ both formally and in your own words.
4. How can we show that two sets equal using $\subseteq$?
5. Define $A \cup B$ and $A \cap B$ in you own words. Then write down the formal definitons.
6. Define $A \setminus B$ both formally and in your own words.
7. Remember the laws for set operations from Theorem 3.4. Which of those laws are similar to some we saw in a previous chapter? Can you prove those using the rules you already know from propositional logic?
8. What is the symbol for the empty set? Name an interesting property of the empty set.
9. Consider the following sets: $ \{ \varnothing \} , \varnothing , \{ \varnothing , \{ \varnothing \} \}, \{\{ \varnothing \}\}$. How many elements do they have? Which ones are subsets of some other? Can you construct one set using a set operation on two others?
10. Define the powerset of a set. How many elements does the powerset of A with |A| = 4 have? What is the powerset of $\varnothing$? What is the powerset of $\{ \varnothing \}$?